ARITHMETIC. (PART 5.) INVOLUTION. 1. If a product consists of equal factors, it is called a power of one of those equal factors, and one of the equal factors is called a root of the product. The power and the root are named according to the number of equal factors in the product. Thus, 3 x 3, or 9, is the second power, or square, of 3; 3 × 3 × 3, or 27, is the third power, or cube, of 3; 3 × 3 × 3 × 3, or 81, is the fourth power of 3. Also, 3 is the second root, or square root, of 9; 3 is the third root, or cube root, of 27; 3 is the fourth root of 81. 2. For the sake of brevity, 3 x 3 is written 3', and read three square, 3 x 3 x 3 is written 3', and read three cube, 3 × 3 × 3 × 3 is written 3*, and read three fourth, and so on. or three exponent four; A number written above and to the right of another number, to show how often the latter number is used as a factor, is called an exponent. Thus, in 3", the number " is the exponent, and shows that 3 is to be used as a factor twelve times; so that 3" is a contraction for 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3. COPYRIGHTED BY INTERNATIONAL TEXTBOOK COMPANY. ENTERED AT STATIONERS' HALL, LONDON In an expression like 3', the exponent shows how often 3 is used as a factor. Hence, if the exponent of a number is unity, the number is used once as a factor; thus, 3' 4' 4, 5' 5. = = = 3, 3. If the side of a square contains 5 inches, the area of the square contains 5 × 5, or 5', square inches. If the edge of a cube contains 5 inches, the volume of the cube contains 5 × 5 × 5, or 5', cubic inches. It is for this reason that 5' and 5' are called the square and cube of 5, respectively. 4. To find any power of a number: EXAMPLE 1.-What is the third power, or cube, of 35? 35 X 35 X 35 EXAMPLE 4.-What is the third power, or cube, of ? SOLUTION. 5. Rule.-I. To raise a whole number or a decimal to any power, use it as a factor as many times as there are units in the exponent. II. To raise a fraction to any power, raise both the numerator and denominator to the power indicated by the exponent. EXAMPLES FOR PRACTICE. Raise the following to the powers indicated: EVOLUTION. DEFINITIONS AND GENERAL REMARKS. 6. Evolution is the reverse of involution. It is the process of finding the root of a number that is considered as a power. 7. The square root of a number is that number which, when used twice as a factor, produces the number. Thus, 2 is the square root of 4, since 2 X 2 (or 21) = 4. 8. The cube root of a number is that number which, when used three times as a factor, produces the number. Thus, 3 is the cube root of 27, since 3 × 3 × 3 (qr 3') = 27. 9. The fourth root of a number is that number which, when used four times as a factor, produces the number. Thus, 9 is the fourth root of 6,561, since 9 x 9 x 9 x9 (or 9') = 6,561. 10. The fifth root of a number is that number which, when used five times as a factor, produces the number. Thus, 7 is the fifth root of 16,807, since 7 X 7 X 7 X 7 X7 (or 7°) 16,807. = 11. The processes of finding squares and cubes and square roots and cube roots are very frequently employed in connection with the solution of problems pertaining to mensuration and engineering. The process of raising a number to some power, the exponent of the number being integral (integral is the adjective for integer; i. e., an integral number is one that does not contain a fraction or decimal) is very simple; but the reverse process, that of finding the roots, is very long and laborious, for which reason tables are generally employed. The tables so used are of two kinds those giving the roots directly and logarithms. While the roots of numbers can be found without the aid of a table, it is not customary to do this except in the case of square root, which is comparatively easy. At the same time it is well to know some general method of finding the roots of numbers, as it might be necessary to find a root when a table was not at hand. For purposes of this Course, a knowledge of how to use a table is all that is necessary. 12. Some idea of the importance of the processes of involution and evolution may be obtained from the following: In finding the area of a square or a circle, it is necessary to square the length of a certain line; conversely, in finding the side of a square or the diameter of a circle that will have a given area it is necessary to find the square root. In finding the volume of a cube or a sphere it is necessary to cube the length of a certain line; conversely, in finding the length of one of the edges of a cube or the diameter of a sphere that will have a given volume, it is necessary to find the cube root. There are many other cases where it is required to extract square root and cube root, but enough has been stated so far to show the importance of the processes. 13. Cube root is not required as often as square root; fourth and fifth roots are required but very seldom, and not at all in connection with the work of this Course. 14. Having shown the necessity of some means of finding the roots of numbers, the manner of using the table following the Examination Questions in this Section will now be explained. But before studying the explanations, certain definitions and properties of numbers must be carefully considered. 15. The radical sign √, when placed before a number, indicates that some root of that number is to be found. The vinculum is almost always used in connection with the radical sign, as shown in Art. 16. 16. The index of the root is a small figure placed over and to the left of the radical sign, to show what root is to be found. |